Vertex connectivity

Each crack intersection point is connected to other crack intersection points. The next logical step is to find out, given a crack intersection point, what are its neighbouring crack intersection points?

Two crack intersection points c₁ and c₂ are neighbours if and only if the condition

M=n (L_c1∩ L_c2) = 4 (2.4)

is satisfied where Mis the total number of elements in the intersection of two labels. This can be seen in figure 2.9 (a). For instance, the crack intersection point formed by regions 1,2 and 4 has three neighbours. The condition in equation 2.4 is tested for all pairs of crack intersection points and a list of neighbours is compiled.

For each crack intersection point c , N_c can be defined as a list of its neighbours. For example, for point A theN_A ={B, C, D}. This list of neighbours is calculated for all crack intersection points and used to make measurements on the crack network. The details of these measurements will be covered in the next chapter.

Summary

This chapter presents the sequence of image processing steps: starting from the experimental procedure to finally acquiring a binary, skeleton image along with a connectivity matrix for crack intersection points.

The experimental procedure for generating crack patterns is described in the first section of the chapter, followed by a brief description about the types of crack patterns observed

1

Figure 2.11: Neighbours algorithm. Point A is at the intersection of three regions -2, 4 and 5 and has three neighbours, points B, C and D. With point B, point A shares two regions- 2 and 4. With point C, it shares two regions- 4 and 5. With point D, it shares region 2 and 5.

Apart from this, the third region, which is shared by all the crack intersection points is the crack pattern skeleton. This algorithm is repeated for all crack intersection points.

with increasing layer height. Two set of plates are used where radial plates 1 and 2 generate crack patterns with radial symmetry. The main control parameter for the experiments is the layer height. The layer height depends on the mass of bentonite used during each trial of the experiments. In a sequence of trials, the bentonite mass is increased by 10 grams and crack patterns are generated.

Images of the crack pattern are captured and processed. The image processing step is needed to ensure the quantification of the crack pattern. Raw images of the crack pattern, which are in jpeg format are converted to 8 bit, then a band-pass filter is applied to them.

Structures upto 40 pixels and down till 3 pixels are filtered. The filtering process removes uneven lighting especially shadows. The filtered image is contrast adjusted to generate a grayscale image. This grayscale image is converted to a binary image, then to a skeleton image.

The second part of image processing involves elimination of spurs and the acquisition of crack intersection points, its neighbours and a connectivity matrix. Spurs are eliminated by converting a skeleton image of a crack pattern to a labelled image, then checking pixel neighbourhood of all the pixels that form a connected crack network. By applying equation 2.2 to the pixel neighbourhood spurs are eliminated. Crack intersection points are also identified in a similar way, where equation 2.3 is use to decide if a point qualifies as a crack intersection point or not. Once the crack intersection points have been identified, its first neighbours are identified and stored as a list.

In the next chapter, all the components- from the binary image to the list of neighbours for each crack intersection points, are used to define measures of a crack pattern

Chapter 3

Analysis of crack patterns

In the previous chapter, the prerequisites for analysis of a crack pattern were developed.

These prerequisites are: a labeled skeleton image of the crack pattern, a list of all the crack intersection points and a list of neighbours of each crack intersection point. Using these, the tools required to quantify these crack patterns are developed in this chapter.

So far, the description of crack patterns has been limited to a qualitative classification of being wavy, ladder-like and isotropic. This chapter presents methods which attempt to quantify the structure of a crack pattern. The methods are presented for images of sinusoidal plates, and unless stated otherwise, also apply to radial plates. These analysis methods are later applied to images of crack patterns for increasing layer heights, the data and the interpretation is presented in the next chapter.

The content of this chapter is roughly divided into two sections, as shown in figure 3.1.

The first section deals with various measures which provide robust methods for quantifying crack patterns. They condense information about the crack pattern into a single number, which can be used, as seen in the next chapter, to show how substrates affect a crack pattern with changing layer height. Three measures for a crack pattern are presented: the orientation of crack intersection point neighbours, the orientation of cracked regions and the orientation of individual cracks. The first order parameter involves measuring angles at crack intersection points. Here angles are measured between the horizontal and~rij, which is the vector between the i^th crack intersection point and its j^th neighbour. The crack angles are measured for neighbours of thei^th crack intersection point using which an angle distribution is generated.

This angle distribution is condensed into a single value. In this section, the methods to calculate angles for both sinusoidal and radially sinusoidal plates are described. Angles are calculated for a crack pattern on sinusoidal and radially sinusoidal plates and the resulting angle distributions are shown. The angles distributions are multiplied with a cos (4θ) function and the value for S_Angles is calculated. This value is used as a measure of order of a crack pattern in the next chapter.

The second measure is the orientation of cracked regions. This parameter involves detect-ing isolated cracked regions, fittdetect-ing them with ellipses, and measurdetect-ing their orientation. The average value of the orientation, SOrt, is used as measure of order. The orientation of each ellipse is a local measure of the order. The required methods to detect the cracked regions

and calculate their orientation are described.

The third measure is the measurement of crack orientation, using methods developed for measuring the orientation of cracked regions, the orientation of crack skeletons is measured and angle distributions are plotted. The angle distributions calculated from the measurement of crack orientation are analyzed in the same way as the angle distributions acquired from measuring angles around crack intersection points. This method is similar to the first measure since both involve calculating the orientation of cracks. This method is a direct measure of the orientation of the cracks; it can be applied to unconnected cracks as well, whereas the first measure requires a crack network in order to be used.

The second set of topics covered in the chapter involve two global methods of quantifying crack patterns- the Manhattan metric and the Minkowski parameters. Other methods which assist in describing the crack pattern quantitatively are- calculation of the crack spacing, determination of crack lengths and determination of number of crack per wavelength of the sinusoidal plate using Fourier methods.

The Manhattan metric approach is an attempt to define a global order parameter. The Manhattan metric provides a measure of the distance between two points on a grid by travers-ing either in thexdirection or they direction, but not a combination of both. Such a metric is used to study geometries known as taxicab geometries. Simple examples of taxicab geome-tries are, traveling in the grid like streets of Manhattan ( after which the metric has been named) or moving on a chessboard. The analogy of the Manhattan metric is used to calculate the distance traveled along the cracks between two crack intersection points. A ratio of the straight line distance between the two crack intersection points and the distance traveled along the crack is used to define a parameters_{M an}. This parameter is calculated for all pairs of crack intersection points which are sorted based on the symmetry of the substrate. The mean ofS_{M an} is used to represent a measure of order of the crack pattern.

Another global parameter that is presented uses Minkowski parameters to quantify the structure of a crack pattern. The Minkowski parameters are defined in the context of a foreground and a background of a spatially varying pattern. The foreground in the current case are the crack intersection points which are plotted as single black pixels on a background of white pixels. For increasing radii of disks that have their origins at these single pixels, the three Minkowski parameters for two dimensions - the area, the boundary and the Euler number are measured. A sample crack pattern is analyzed using the Minkowski parameters and results are plotted for increasing radius of disks. The Minkowski parameters, unlike the previous order parameters are not presented as a single value that provides a measure of the crack pattern. Instead three plots, one for each Minkowski parameter, are generated and shown.

The Minkowski parameters are followed by methods which describe the measurement of the crack spacing of a crack pattern. The crack spacing is measured in order to detect correlations between the substrate structure and the geometry of the crack pattern. Further-more, the crack spacing of an isotropic crack pattern is known. Hence, the question asked is:

will crack spacing at large layer heights approach the crack spacing for previously measured

isotropic crack patterns? The crack spacing is measured in two ways. One involves using the line dropping method where lines are plotted on a skeleton image of the crack pattern and the distances between the crack and line intersections are measured and averaged. The second method involves measuring the areas of cracked regions. The side of a square of the same area is also used as a measure of the crack spacing.

The lengths of cracks are measured in order to calculate and study the change in the distribution of crack length. The length of each crack is measured using the same methods developed in the section where orientation of cracks are measured.

The final section of this chapter is one where Fourier methods are used to analyse the crack pattern generated on a sinusoidal substrate. This is necessary in order to determine for a given layer height, and answer of how many cracks there are per wavelength in a mature crack pattern. The methods to calculate the spectral power of the crack density of a sample crack pattern are described using the example of a ladder-like crack pattern. The spectral power is plotted against the wave-number, which represents the number of wavelengths in the dimensions of the sinusoidal plate.

Figure 3.1: An outline of the topics in the chapter. The chapter can be divided into two sections. The measures aim to assign a single number to a crack pattern which quantifies the effect of the substrate on the crack pattern, while the second section contains various methods to describe the geometry and topology of a crack pattern.

3.1 Crack angles distribution

The distribution of crack angles serves as a useful tool for comparison of crack patterns. In this section the methods to acquire the angle distribution and the subsequent order parameters are presented. A skeleton image of the crack pattern is used to measure the crack angles. In the skeleton image, after identifying each crack intersection point, the crack angles are calculated between the horizontal unit vector ˆx = [1 0] and~r_ij, which is the vector that connects thei^th crack intersection point with itsj^th neighbours. Take the example in figure 3.2. Let ˆx be the horizontal unit vector that lies on the dashed green line. Then, the angle θ₁ is defined by

-cosθ₁ =~r₁·ˆx

|~r1| . (3.1)

Similarly θ2 and θ3 can be calculated. Figure 3.2 (a) is an image of the crack pattern, and figure 3.2 (c) is the corresponding angle distribution.

Figure 3.2: Angle distribution of a crack pattern over sinusoidal substrate. (a) depicts the the three neighbouring crack intersection points to thei^th crack intersection point. The red vectors join the crack intersection point to its neighbours, an example of this it the vector~r₁. The angleθ₁ is defined between the 1^st crack intersection point and the green vector which is ˆ

x = [1 0], it is perpendicular to the peak. (b) represents the full crack pattern from which the crack intersection point in (a) is selected. The red points are the crack intersection points.

Figure (c) is an angle distribution of the all the measured angles. This angle distribution goes fromθ= 0^◦ toθ= 180^◦ and has 36 bins.

For the radial crack pattern in (figure 3.3), a crack intersection point and its neighbours are shown in figure 3.3 (a). The equivalent of the horizontal vector ˆx is the radial unit vectorˆr_k which is parallel to the vector that connects the center of the image and the i^th crack intersection point. The angleθ1 is calculated

using-cosθ₁=~r1·ˆrk

|~r1| . (3.2)

For all crack intersection points, angles between~r_ij and the direction of the substrate ˆx orrˆ_k (depending on the system’s symmetry) are calculated and plotted as a histogram. The

Figure 3.3: Angle distribution of a crack pattern over radial sinusoidal substrate. (a) shows the neighbours of thei^th crack intersection point which is part of a crack pattern generated on a radially sinusoidal plate. The blue vector connects the center of image and thei^thcrack intersection point. The vectorˆr_k is the vector perpendicular to circle lying atop the peak.

This vector is parallel to the blue vector. Figure (b) is a radially symmetric crack pattern generated over the radially sinusoidal substrate. The red points are crack intersection points.

Figure (c) is angle distribution for the image in (b). The majority of the angles lie around 0^◦,90^◦ and 180^◦.

where P⁰ (θ_i) is the probability distribution for the angles. N is the total number of bins for the angle distribution. θi refers to the center value of thei^thbin. In general,N = 36 bins are used to generate the angle distributions, henceθ₁= 2.5^◦, θ₂= 7.5^◦ and so on. M is the total number of angles measured over all crack intersection points. The probability distribution is normalized by dividing the total count of angles in each bin by M such that,

1 intersection points and hence varying number of measured angles, applying this normalization condition scales the angle distribution such that it falls between 0 and 1 allowing for direct comparison between two crack patterns.

Sample angle distributions are presented in figure 3.2 (c) figure 3.3 (c). For the crack pattern in figure 3.3 (b), the angle distribution has peaks atθ= 0^◦,90^◦,180^◦ (figure 3.3 (c)).

Once an angle distribution has been generated, it is multiplied by a cos 4θ function and summed overθ to give the following order parameter,

SAngles (θ) =

N

X

i=1

[cos (4θi) P (θi))], (3.5) whereN is the total number of bins ofθ.

From equation 3.5, S_Angles will lie between 1 and -1. The S_Angles = 1 case represents cracks that lie parallel to the substrate, while SAngles = 0 case represents an isotropic crack pattern, and S_Angles =−1 caseS represents cracks that lie at 45^◦ from the substrate.

The S_Anglesparameter requires prior information about the symmetry of the crack pattern.

For example, in case of the radial plates, without knowing about the radial symmetry of the substrate, it is not possible to determine whether equation 3.1 or equation 3.2 should be used to calculate the angle distribution. However, once the symmetry has been determined, S_Angles allows for comparisons between crack patterns (presented in the next chapter). It simplifies an angle distribution down to a single number that can be assigned to each crack pattern.